Chapter 11

(a) Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the \(x y\) -term. (c) Sketch the graph. $$25 x^{2}-120 x y+144 y^{2}-156 x-65 y=0$$

15–22 (a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. $$r=\frac{5}{2-3 \sin \theta}$$

Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$ y^{2}=4(x+2 y) $$

Use a graphing device to graph the parabola. $$x^{2}=-8 y$$

\begin{array}{l}{1-22 \text { a pair of parametric equations is given. }} \\\ {\text { (a) Sketch the curve represented by the parametric equations. }} \\\ {\text { (b) Find a rectangular-coordinate equation for the curve by }} \\\ {\text { eliminating the parameter. }}\end{array} $$ x=\cos ^{2} t, \quad y=\sin ^{2} t $$

(a) Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the \(x y\) -term. (c) Sketch the graph. $$\sqrt{3} x^{2}+3 x y=3$$

15–22 (a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. $$r=\frac{7}{2-5 \sin \theta}$$

Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$ x^{2}-4 y^{2}-2 x+16 y=20 $$

Use a graphing device to graph the parabola. $$y^{2}=-\frac{1}{3} x$$

\begin{array}{l}{1-22 \text { a pair of parametric equations is given. }} \\\ {\text { (a) Sketch the curve represented by the parametric equations. }} \\\ {\text { (b) Find a rectangular-coordinate equation for the curve by }} \\\ {\text { eliminating the parameter. }}\end{array} $$ x=\cos ^{3} t, \quad y=\sin ^{3} t, \quad 0 \leq t \leq 2 \pi $$

(a) Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the \(x y\) -term. (c) Sketch the graph. $$153 x^{2}+192 x y+97 y^{2}=225$$

15–22 (a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. $$r=\frac{8}{3+\cos \theta}$$

Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$ x^{2}+6 x+12 y+9=0 $$

Use a graphing device to graph the parabola. $$8 y^{2}=x$$

Find parametric equations for the line with the given properties. Slope \(\frac{1}{2},\) passing through \((4,-1)\)

(a) Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the \(x y\) -term. (c) Sketch the graph. $$2 \sqrt{3} x^{2}-6 x y+\sqrt{3} x+3 y=0$$

(a) Find the eccentricity and directrix of the conic \(r=1 /(4-3 \cos \theta)\) and graph the conic and its directrix. (b) If this conic is rotated about the origin through an angle \(\pi / 3,\) write the resulting equation and draw its graph.

Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$ 4 x^{2}+25 y^{2}-24 x+250 y+561=0 $$

Use a graphing device to graph the hyperbola. \(x^{2}-2 y^{2}=8\)

Use a graphing device to graph the parabola. $$4 x+y^{2}=0$$

Find parametric equations for the line with the given properties. Slope \(-2,\) passing through \((-10,-20)\)

(a) Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the \(x y\) -term. (c) Sketch the graph. $$9 x^{2}-24 x y+16 y^{2}=100(x-y-1)$$

Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$ 2 x^{2}+y^{2}=2 y+1 $$

Use a graphing device to graph the hyperbola. \(3 y^{2}-4 x^{2}=24\)

Use a graphing device to graph the parabola. $$x-2 y^{2}=0$$

Find parametric equations for the line with the given properties. Passing through \((6,7)\) and \((7,8)\)

Use a graphing device to graph the ellipse. $$ \frac{x^{2}}{25}+\frac{y^{2}}{20}=1 $$

(a) Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the \(x y\) -term. (c) Sketch the graph. $$52 x^{2}+72 x y+73 y^{2}=40 x-30 y+75$$

Graph the conics \(r=e /(1-e \cos \theta)\) with \(e=0.4,0.6,0.8\) and 1.0 on a common screen. How does the value of \(e\) affect the shape of the curve?

Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$ 16 x^{2}-9 y^{2}-96 x+288=0 $$

Use a graphing device to graph the hyperbola. \(\frac{y^{2}}{2}-\frac{x^{2}}{6}=1\)

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Focus \(F(0,2)\)

Find parametric equations for the line with the given properties. Passing through \((12,7)\) and the origin

Use a graphing device to graph the ellipse. $$ x^{2}+\frac{y^{2}}{12}=1 $$

(a) Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the \(x y\) -term. (c) Sketch the graph. $$(7 x+24 y)^{2}=600 x-175 y+25$$

(a) Graph the conics $$r=\frac{e d}{1+e \sin \theta}$$ for \(e=1\) and various values of \(d\) . How does the value of \(d\) affect the shape of the conic? (b) Graph these conics for \(d=1\) and various values of \(e\) . How does the value of \(e\) affect the shape of the conic?

Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$ 4 x^{2}-4 x-8 y+9=0 $$

Use a graphing device to graph the hyperbola. \(\frac{x^{2}}{100}-\frac{y^{2}}{64}=1\)

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Focus \(F\left(0,-\frac{1}{2}\right)\)

Find parametric equations for the circle \(x^{2}+y^{2}=a^{2}\)

Use a graphing device to graph the ellipse. $$ 6 x^{2}+y^{2}=36 $$

(a) Use the discriminant to identify the conic. (b) Confirm your answer by graphing the conic using a graphing device. $$2 x^{2}-4 x y+2 y^{2}-5 x-5=0$$

Orbit of the Earth The polar equation of an ellipse can be expressed in terms of its eccentricity \(e\) and the length \(a\) of its major axis. (a) Show that the polar equation of an ellipse with directrix \(x=-d\) can be written in the form $$r=\frac{a\left(1-e^{2}\right)}{1-e \cos \theta}$$ [Hint: Use the relation \(a^{2}=e^{2} d^{2} /\left(1-e^{2}\right)^{2}\) given in the proof on page \(843 . ]\) (b) Find an approximate polar equation for the elliptical orbit of the earth around the sun (at one focus) given that the eccentricity is about 0.017 and the length of the major axis is about \(2.99 \times 10^{8} \mathrm{km} .\)

Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$ x^{2}+16=4\left(y^{2}+2 x\right) $$

Find an equation for the hyperbola that satisfies the given conditions. Foci \(( \pm 5,0),\) vertices \(( \pm 3,0)\)

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Focus \(F(-8,0)\)

Find parametric equations for the ellipse $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 $$

Use a graphing device to graph the ellipse. $$ x^{2}+2 y^{2}=8 $$

(a) Use the discriminant to identify the conic. (b) Confirm your answer by graphing the conic using a graphing device. $$x^{2}-2 x y+3 y^{2}=8$$

Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$ x^{2}-y^{2}=10(x-y)+1 $$